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Tetris Is Hard with Just One Piece Type

Authors MIT Hardness Group, Josh Brunner , Erik D. Demaine , Della Hendrickson , Jeffery Li

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • complexity
  • hardness
  • video games
  • counting

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Abstract

We analyze the computational complexity of Tetris clearing (determining whether the player can clear an initial board using a given sequence of pieces) and survival (determining whether the player can avoid losing before placing all the given pieces in an initial board) when restricted to a single polyomino piece type. We prove, for any tetromino piece type P except for O, the NP-hardness of Tetris clearing and survival under the standard Super Rotation System (SRS), even when the input sequence consists of only a specified number of P pieces. These surprising results disprove a 23-year-old conjecture on the computational complexity of Tetris with only I pieces (although our result is only for a specific rotation system). As a corollary, we prove the NP-hardness of Tetris clearing when the sequence of pieces has to be able to be generated from a 7k-bag randomizer for any positive integer k ≥ 1. On the positive side, we give polynomial-time algorithms for Tetris clearing and survival when the input sequence consists of only dominoes, assuming a particular rotation model, solving a version of a 9-year-old open problem. Along the way, we give polynomial-time algorithms for Tetris clearing and survival with 1 × k pieces (for any fixed k), provided the top k-1 rows are initially empty, showing that our I NP-hardness result needs to have filled cells in the top three rows.

Cite As Get BibTex

MIT Hardness Group, Josh Brunner, Erik D. Demaine, Della Hendrickson, and Jeffery Li. Tetris Is Hard with Just One Piece Type. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 32:1-32:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.32

Author Details

MIT Hardness Group
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Josh Brunner
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Erik D. Demaine
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Della Hendrickson
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Jeffery Li
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA

Acknowledgements

This paper was initiated during open problem solving in the MIT class on Algorithmic Lower Bounds: Fun with Hardness Proofs (6.5440) taught by Erik Demaine in Fall 2023. We thank the other participants of that class - specifically Eugeniya Artemova, Cecilia Chen, Lily Chung, Jenny Diomidova, Holden Hall, Jonathan Li, Jayson Lynch, Frederick Stock, and Ethan Zhou - for helpful discussions and providing an inspiring atmosphere.

References

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